Formulated over 2,000 years ago, the Pythagoras theorem is much more than a mathematical formula. It is a real timeless pillar of Euclidean geometry. This theorem is used to calculate the length on the longest side of a rectangle triangle, what is called hypotenuse.
A trigonometric demonstration to solve Pythagoras' theorem
In 2022, two American high school girls, Ne'kiya Jackson and Calcea Johnson, proposed a new approach to solve the Pythagoras theorem. Their demonstration is all the more remarkable since it only uses trigonometry, which was thought to be impossible to do so far.
Geometric and algebraic methods have often been used to prove the Pythagoras theorem, but we had not yet seen an exclusively trigonometric demonstration. And for good reason, trigonometric functions like sinus and cosine are intrinsically linked to this theorem.
So how did the teenagers do? They had a real idea of genius. Instead of depending on the Pythagoras theorem in their demonstration, they relied on fundamental principles of geometry such as the properties of angles and proportions.
They thus avoided circular reasoning using more basic and more direct proof.
A genius approach using trigonometry
Concretely, they first built rectangle triangles and other figures of geometry using relations of proportionality between the angles and the sides.
They then defined trigonometric functions (sinus and cosine) from the properties of the angles and proportions in their already built triangles. This made it possible to establish relations between the lengths of the sides of the rectangle triangles.
The use of trigonometric identities has made it possible to simplify these relationships. For example, the identity that connects the sinus and the cosine is “sin² (x) + cos² (x) = 1”. After several stages of calculation, the young girls finally arrived at the equation of the Pythagoras theorem: “A² + B² = C²”.
They have thus proven that it was possible to demonstrate the Pythagoras theorem without ever using it only once in the demonstration.
An approach immediately recognized by the scientific community
After four years of research, Ne'kiya Jackson and Calcea Johnson presented their work at the annual conference of the Mathematical Association of America in Atlanta, in March 2023.
The scientific community immediately recognized their approach, which earned them the publication of their work in the magazine American Mathematical Monthly a few months later.
Moreover, their article is not limited to a single demonstration of the Pythagoras theorem. They proposed several methods. One of them even generates five other distinct demonstrations.
The new student approach could allow new applications in pure and applied mathematics to see the light of day. Other researchers could actually find alternative approaches to already established theorems.
This demonstration of the Pythagoras theorem is a source of inspiration for young people
Ne'kiya Jackson and Calcea Johnson hope that their work will inspire youth. “This shows that even students can contribute to the progress of knowledge,” said Calcea Johnson who is a student in environmental engineering today at the State University of Louisiana.
Ne'kiya Jackson is a pharmacy student at the University of Xavier de Louisiana. “We want to show that with passion and perseverance, anything is possible,” she confides.
New perspectives for the mathematics of tomorrow
Their demonstration is brilliant proof that mathematics is a constantly evolving living science. It is always possible to make new discoveries, even on subjects as old as the theorem of Pythagoras.
In the long term, their approach could have a significant impact on future work in geometry, trigonometry, applied sciences and even in artificial intelligence.
For example, we can expect more efficient algorithms to be developed thanks to new approaches in geometry and trigonometry. The potential repercussions of their demonstration is undoubtedly a source of inspiration for many researchers around the world.
With an unwavering passion for local news, Christopher leads our editorial team with integrity and dedication. With over 20 years’ experience, he is the backbone of Wouldsayso, ensuring that we stay true to our mission to inform.
